Download - Factoring
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Factoring
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Factoring by Greatest Common Factor (GCF)
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Factoring by removing the GCF “undoes” the multiplication. (It’s the distributive property in reverse.)
2x2 – 4
2x2 – (2 2)
2(x2 – 2)
Check 2(x2 – 2)
2x2 – 4
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Example
8x3 – 4x2 – 16x
2x2(4x) – x(4x) – 4(4x)
4x(2x2 – x – 4)
4x(2x2 – x – 4)
8x3 – 4x2 – 16x
Check
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Factoring ax2+bx+c
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2x bx c
To factor, find two numbers that:
Add to b Multiply To c
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Example
Factor x2 + 15x + 36. Check your answer.
(x + )(x + )
(x + 1)(x + 36) = x2 + 37x + 36
(x + 2)(x + 18) = x2 + 20x + 36
(x + 3)(x + 12) = x2 + 15x + 36
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Example
Factor each trinomial. Check your answer.
x2 + 10x + 24
(x + )(x + )
(x + 4)(x + 6) = x2 + 10x + 24
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Factor each trinomial.
x2 + x – 20
(x + )(x + )
(x – 4)(x + 5)
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Remember to look for a GCF first!
3x2 + 18x – 21
3(x + )(x + )
3(x – 1)(x + 7)
3(x2 + 6x – 7)
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Factor each trinomial.
x2 – 11xy + 30y2
(x – 5y)(x – 6y)
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A trinomial is a perfect square if:
• The first and last terms are perfect squares.
• The middle term is double the product of
the square roots.
9x2 + 12x + 4
3x 3x 2(3x 2) 2 2 • • •
REMEMBER THE X!
Perfect Square Trinomials
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Factor.
81x2 + 90x + 25
Example
(9x)( 9x) (5)( 5)
The middle term = 2(9x)(5), so this is a perfect square trinomial
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Factor.
Example
281 90 25
(9 )(9 )
x x
x x
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Factor.
Example
281 90 25
(9 5 )(9 5)
x x
x x
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A polynomial is a difference of two squares if:
•There are two terms, one subtracted from the other.
• Both terms are perfect squares.
4x2 – 9
2x 2x 3 3
For variables: All even powers are perfect squares
Difference of Two Squares
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Example
249 100x
(7x)( 7x) (10)( 10)
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Example
249 100
(7 10)(7 10)
x
x x
Center term cancels out, so use opposite signs.
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Factoring ax2 + bx + c
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The “box” method
22x
6
First term
Last term
Include the signs!!!
22 7 6x x
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Multiply a and c. (In this case, that would be 2 x 6 = 12) Put the
factors of “ac” that add up to “b” in the other squares, with their signs and an “x”. Order does not matter.
22x
6
One factor
The other factor
3x
4x
22 7 6x x
4 and 3 are factors of 12 that add up to 7, so they go in the empty spaces. Add an x because they represent the middle term.
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Factor the GCF out of each row or
column. Use the signs from the closest term.
22x
63x
4xGCF is +2x
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Factor the GCF out of each row or
column. Use the signs from the closest term.
22x
63x
4x+2x
GCF is +3
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Factor the GCF out of each row or
column. Use the signs from the closest term.
22x
63x
4x
GCF is +x
+2x
+3
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Factor the GCF out of each row or
column. Use the signs from the closest term.
22x
63x
4x
+x GCF =+2
+2x
+3
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The “outside” factors combine to factor the quadratic.
22x
63x
4x
+x +2
+2x
+3
(2x+3)(x+2)
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You can also use trial and error. Determine the possible factors for the first and last terms, and then keep trying combinations until you find the
one that works.
23 17 10 x x
First term: 3x and x are the only possible factors Last term: Factors are 1 and 10 or 2 and 5
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2(3 1)( 10) 3 31 10 x x x x
2(3 10)( 1) 3 13 10 x x x x
2(3 5)( 2) 3 11 10 x x x x
2(3 2)( 5) 3 17 10 x x x x
Only this combination works.
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Factoring by Grouping
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Factoring by grouping: If you have four terms – make 2 groups of 2 and factor out the GCF from each. MUST be used on 4 terms CAN be used on 3.
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Example
Factor each polynomial by grouping. Check your answer.
6h4 – 4h3 + 12h – 8
(6h4 – 4h3) + (12h – 8)
2h3(3h – 2) + 4(3h – 2)
2h3(3h – 2) + 4(3h – 2)
(3h – 2)(2h3 + 4)
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Factor each polynomial by grouping. Check your answer.
Check (3h – 2)(2h3 + 4)
3h(2h3) + 3h(4) – 2(2h3) – 2(4)
6h4 + 12h – 4h3 – 8
6h4 – 4h3 + 12h – 8
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Example
Factor each polynomial by grouping. Check your answer.
5y4 – 15y3 + y2 – 3y
(5y4 – 15y3) + (y2 – 3y)
5y3(y – 3) + y(y – 3)
5y3(y – 3) + y(y – 3)
(y – 3)(5y3 + y)
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Factor each polynomial by grouping. Check your answer.
5y4 – 15y3 + y2 – 3y
Check (y – 3)(5y3 + y)
y(5y3) + y(y) – 3(5y3) – 3(y)
5y4 + y2 – 15y3 – 3y
5y4 – 15y3 + y2 – 3y
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You can use factoring by grouping on trinomials.
23 11 10x x
Split the 11x into two terms (coefficients should multiply to 30, because 3x10=30)
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23 6 5 10x x x
Write the terms in whichever order will allow you to group.
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2(3 6 ) (5 10)
3 ( 2) 5( 2)
(3 5)( 2)
x x x
x x x
x x